Abstract group 51840.bb: $F_9:S_6$

• Label: 51840.bb
• Order: \( 2^{7} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$F_9:S_6$
Order:	\(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$\PSU(3,2):C_2.A_6.C_2^2$, of order \(207360\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $A_6$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	40
Elements	1	819	728	8748	144	10440	4896	1296	12960	1152	2592	2880	5184	51840
Conjugacy classes	1	5	5	9	1	9	6	1	6	1	2	4	4	54
Divisions	1	5	5	9	1	9	3	1	6	1	1	2	1	45
Autjugacy classes	1	4	3	8	1	5	6	1	4	1	1	2	2	39

Dimension	1	2	4	5	8	9	10	16	18	20	32	36	40	64	72	80	128
Irr. complex chars.	4	3	0	8	2	4	10	8	3	3	0	0	4	0	2	2	1	54
Irr. rational chars.	4	1	1	8	2	4	6	2	1	3	1	1	5	1	2	2	1	45

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$2544$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	40	40	40
Arbitrary	13	13	13

Constructions

Permutation group:	Degree $15$ $\langle(7,12)(8,11)(9,10)(13,15), (7,8,9)(10,11,12)(13,14,15), (7,8,12,11)(9,13,10,15) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$F_9$ $\,\rtimes\,$ $S_6$	$(F_9\times A_6)$ $\,\rtimes\,$ $C_2$	$A_6$ $\,\rtimes\,$ $(F_9:C_2)$	$C_3^2$ $\,\rtimes\,$ $(C_8:S_6)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3\times A_6)$ . $D_4$	$(C_3:S_3)$ . $(C_4:S_6)$	$(C_3^2:C_4)$ . $(C_2\times S_6)$	$(C_3^2:C_4\times A_6)$ . $C_2^2$	all 5

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 320286 subgroups in 1156 conjugacy classes, 13 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $F_9:S_6$
Commutator:	$G' \simeq$ $C_3^2:C_4\times A_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $F_9:S_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2$	$G/\operatorname{Fit} \simeq$ $C_8:S_6$
Radical:	$R \simeq$ $F_9$	$G/R \simeq$ $S_6$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times A_6$	$G/\operatorname{soc} \simeq$ $\SD_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 51840: $F_9:S_6$

Order 25920: $A_6:\SOPlus(4,2)$, $F_9\times A_6$, $A_6:\PSU(3,2)$

Order 12960: $A_6:S_3^2$, $(C_3^2\times A_6):C_4$, $C_3^2:C_4\times A_6$

Order 8640: $F_9:S_5$ x 2

Order 6480: $C_3:S_3\times A_6$, $C_3^2:S_6$

Order 5760: $C_8:S_6$

Order 5184: $F_9:\SOPlus(4,2)$

Order 4320: $A_5\times F_9$ x 2, $A_5:\PSU(3,2)$ x 2, $A_5:\SOPlus(4,2)$ x 2, $S_3\times S_6$

Order 3456: $F_9:C_2\times S_4$ x 2

Order 3240: $C_3^2\times A_6$

Order 2880: $C_8\times A_6$, $A_6:Q_8$, $C_4:S_6$

Order 2592: $F_9:S_3^2$ x 2, $C_3^4:(C_2\times \SD_{16})$ x 2, $C_3^3:C_{12}:Q_8$, $C_3^2:C_4\times F_9$, $C_3^3:C_{12}:D_4$

Order 2160: $C_3^2:C_4\times A_5$ x 2, $A_5:S_3^2$ x 2, $(C_3\times \GL(2,4)):C_4$ x 2, $C_3:S_6$, $S_3\times A_6$, $C_3\times S_6$

Order 1728: $S_4\times \PSU(3,2)$ x 2, $D_6^2:D_6$ x 2, $A_4\times F_9:C_2$ x 2, $\PSU(3,2):S_4$ x 2, $\SOPlus(4,2).S_4$ x 2, $F_9:S_4$ x 2, $S_4\times F_9$ x 2

Order 1440: $F_9:F_5$, $C_2\times S_6$, $C_4\times A_6$, $A_6:C_4$

Order 1296: $C_3^4:\SD_{16}$ x 4, $C_3^3:(S_3\times D_4)$ x 3, $C_3^3:(S_3\times Q_8)$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:(C_2\times D_4)$, $C_3^2:(C_2\times F_9)$, $C_3:S_3\times F_9$, $C_3^2\wr C_2.D_4$, $C_3^2\wr C_2.D_4$, $C_3^4:C_4^2$

Order 1152: $D_4\times F_9:C_2$

Order 1080: $\GL(2,4):S_3$ x 2, $C_3^2:S_5$ x 2, $C_3\times A_6$

Order 960: $C_8:S_5$ x 2

Order 864: $C_3^2:C_4\times S_4$ x 4, $A_4:\PSU(3,2)$ x 4, $S_4\times S_3^2$ x 2, $A_4\times \PSU(3,2)$ x 2, $D_6^2:C_6$ x 2, $S_3^2:S_4$ x 2, $C_6^2:D_{12}$ x 2, $A_4\times F_9$ x 2, $A_4:F_9$ x 2, $F_9:D_6$ x 2

Order 720: $S_3\times S_5$ x 2, $D_5\times F_9$, $D_5.\PSU(3,2)$, $D_5.\SOPlus(4,2)$, $C_2\times A_6$, $S_6$

Order 648: $C_3^4:D_4$ x 3, $C_3^3:D_{12}$ x 3, $C_3:S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^4:Q_8$ x 2, $C_3^4:D_4$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^2:F_9$ x 2, $C_3^4:(C_2\times C_4)$, $C_3^2:F_9$, $C_3^2\times F_9$

Order 576: $F_9:C_2^3$ x 2, $C_6^2:\SD_{16}$ x 2, $C_6^2:\SD_{16}$ x 2, $F_9:D_4$ x 2, $D_4\times \PSU(3,2)$, $C_3^2:D_4^2$, $D_4\times F_9$, $\PSU(3,2):D_4$, $(C_3\times C_{12}):\SD_{16}$, $F_9:D_4$, $C_4\times F_9:C_2$, $C_8:\SOPlus(4,2)$

Order 540: $C_3\times \GL(2,4)$ x 2

Order 480: $A_5:Q_8$ x 2, $C_4:S_5$ x 2, $C_8\times A_5$ x 2

Order 432: $C_6^2:C_{12}$ x 4, $(C_3^2\times A_4):C_4$ x 4, $S_3^3:C_2$ x 3, $A_4\times S_3^2$ x 2, $A_4:S_3^2$ x 2, $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $S_3\times \PSU(3,2)$ x 2, $F_9:S_3$ x 2, $C_3^3:\SD_{16}$ x 2, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2, $S_3\times F_9$ x 2

Order 384: $\SD_{16}\times S_4$ x 2

Order 360: $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $C_3\times S_5$ x 2, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $D_5.S_3^2$, $C_5:F_9$, $C_5\times F_9$, $A_6$

Order 324: $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^4:C_4$ x 2, $C_3^4:C_4$ x 2, $C_3\wr C_4$ x 2, $C_3^3:C_{12}$ x 2

Order 288: $F_9:C_2^2$ x 9, $C_6^2:Q_8$ x 4, $\PSU(3,2):C_4$ x 3, $\SOPlus(4,2):C_4$ x 3, $C_3^2:C_4\times D_4$ x 2, $C_4:\PSU(3,2)$ x 2, $C_6^2:D_4$ x 2, $D_6\wr C_2$ x 2, $C_4:\SOPlus(4,2)$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_6^2:C_8$ x 2, $C_{24}:D_6$ x 2, $C_6^2:Q_8$ x 2, $C_6^2:D_4$ x 2, $C_2^2\times F_9$ x 2, $D_6\times S_4$ x 2, $D_4\times S_3^2$, $C_4\times \PSU(3,2)$, $S_3^2:D_4$, $C_4\times \SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $C_4:F_9$, $C_4\times F_9$, $F_9:C_4$, $C_3^2:C_4\times C_8$

Order 240: $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $C_2\times S_5$ x 2

Order 216: $C_3^2:C_4\times S_3$ x 5, $S_3^3$ x 4, $C_3^3:Q_8$ x 4, $S_3^2:S_3$ x 4, $C_3^2:D_{12}$ x 3, $S_3^2:C_6$ x 3, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_3^3:Q_8$ x 2, $C_3:F_9$ x 2, $C_3\times F_9$ x 2

Order 192: $Q_8:S_4$ x 2, $A_4:\SD_{16}$ x 2, $C_8:S_4$ x 2, $C_8\times S_4$ x 2, $Q_8\times S_4$ x 2, $D_4\times S_4$ x 2, $A_4\times \SD_{16}$ x 2

Order 180: $\GL(2,4)$ x 2, $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_{15}:C_{12}$

Order 162: $C_3^2\wr C_2$ x 4, $C_3^3:S_3$

Order 160: $C_{40}:C_4$

Order 144: $S_3\times S_4$ x 10, $S_3^2:C_2^2$ x 8, $F_9:C_2$ x 8, $C_2\times \PSU(3,2)$ x 5, $C_2.\PSU(3,2)$ x 5, $C_6^2:C_4$ x 4, $C_2\times F_9$ x 4, $C_6^2:C_4$ x 4, $C_3^2:\SD_{16}$ x 4, $C_{12}:D_6$ x 3, $C_3^2:C_4^2$ x 3, $C_2.\SOPlus(4,2)$ x 3, $S_3^2:C_4$ x 3, $C_{24}:C_6$ x 2, $C_{12}.D_6$ x 2, $D_6^2$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $D_6:D_6$ x 2, $D_6:D_6$ x 2, $C_{12}.D_6$ x 2, $(C_3\times C_{12}):C_4$ x 2, $C_{24}:S_3$, $C_{12}:D_6$, $C_{12}:D_6$, $S_3\times D_{12}$, $C_4\times S_3^2$, $C_3^2:(C_2\times C_8)$

Order 128: $D_4\times \SD_{16}$

Order 120: $C_2\times A_5$ x 2, $S_5$ x 2, $S_3\times F_5$

Order 108: $C_3:S_3^2$ x 9, $C_3\times S_3^2$ x 8, $C_3^3:C_4$ x 5, $C_3^2:C_{12}$ x 5, $C_3:S_3^2$ x 4, $C_3^2\times A_4$ x 2

Order 96: $C_4\times S_4$ x 4, $A_4:Q_8$ x 4, $C_8\times A_4$ x 2, $A_4:C_8$ x 2, $C_2^2\times S_4$ x 2, $Q_8\times A_4$ x 2, $D_4\times A_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_4:S_4$ x 2, $S_3\times \SD_{16}$ x 2, $D_4\times D_6$

Order 90: $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 81: $C_3^4$

Order 80: $C_{20}:C_4$ x 2, $C_8\times D_5$

Order 72: $\SOPlus(4,2)$ x 12, $C_2\times C_3^2:C_4$ x 10, $S_3\times D_6$ x 9, $S_3\times A_4$ x 6, $C_3:S_4$ x 6, $C_3\times S_4$ x 6, $\PSU(3,2)$ x 5, $F_9$ x 3, $C_3\times D_{12}$ x 3, $D_6:S_3$ x 3, $C_6.D_6$ x 3, $C_6:D_6$ x 2, $C_6\times D_6$ x 2, $C_6\times A_4$ x 2, $C_6^2:C_2$ x 2, $C_{12}:S_3$ x 2, $C_6\wr C_2$ x 2, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $C_3:D_{12}$ x 2, $C_3^2:C_8$ x 2, $D_4\times C_3^2$, $C_3:D_{12}$, $S_3\times C_{12}$, $C_6.D_6$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_2^2\times \SD_{16}$ x 2, $C_8:D_4$ x 2, $C_2^2:\SD_{16}$ x 2, $Q_8:D_4$ x 2, $D_4\times Q_8$, $D_4^2$, $C_8:D_4$, $C_4:\SD_{16}$, $Q_8:D_4$, $C_4\times \SD_{16}$, $C_8\times D_4$

Order 60: $A_5$ x 2, $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$

Order 54: $C_3^2:C_6$ x 9, $S_3\times C_3^2$ x 9, $C_3^2:S_3$ x 5

Order 48: $C_2\times S_4$ x 10, $S_3\times D_4$ x 10, $C_4\times A_4$ x 4, $A_4:C_4$ x 4, $C_{24}:C_2$ x 2, $C_2^2\times D_6$ x 2, $C_2^2\times A_4$ x 2, $C_6:D_4$ x 2, $S_3\times Q_8$ x 2, $S_3\times C_8$ x 2, $C_3\times \SD_{16}$ x 2, $Q_8:S_3$ x 2, $C_3:\SD_{16}$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$, $C_{40}$, $C_5:C_8$

Order 36: $S_3^2$ x 25, $C_3^2:C_4$ x 12, $C_6\times S_3$ x 9, $C_6:S_3$ x 6, $C_3\times A_4$ x 4, $C_3:C_{12}$ x 3, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_6^2$ x 2

Order 32: $C_2\times \SD_{16}$ x 9, $C_2^2:Q_8$ x 4, $D_4:C_4$ x 3, $C_2^2\times D_4$ x 3, $C_4:D_4$ x 3, $C_4\times D_4$ x 3, $Q_8:C_4$ x 3, $C_2^2:C_8$ x 2, $C_2^2\times Q_8$ x 2, $C_2^2\times C_8$ x 2, $C_4:Q_8$ x 2, $C_2^2\wr C_2$ x 2, $C_4:D_4$, $C_4\times C_8$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$ x 5

Order 24: $C_2\times D_6$ x 13, $C_3:D_4$ x 10, $S_4$ x 10, $C_4\times S_3$ x 8, $D_{12}$ x 6, $C_2\times A_4$ x 6, $C_3\times D_4$ x 6, $C_3:Q_8$ x 4, $C_{24}$ x 2, $C_2^2\times C_6$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2, $C_2\times C_{12}$, $C_6:C_4$

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 24, $C_3:S_3$ x 15, $C_3\times C_6$ x 4

Order 16: $C_2\times D_4$ x 14, $\SD_{16}$ x 8, $C_4:C_4$ x 8, $C_2^2:C_4$ x 7, $C_2\times Q_8$ x 5, $C_2^2\times C_4$ x 5, $C_2\times C_8$ x 4, $C_4^2$ x 2, $C_2^4$ x 2

Order 15: $C_{15}$

Order 12: $D_6$ x 22, $C_2\times C_6$ x 7, $C_{12}$ x 6, $C_3:C_4$ x 6, $A_4$ x 4

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 11

Order 8: $D_4$ x 16, $C_2\times C_4$ x 12, $C_2^3$ x 9, $Q_8$ x 5, $C_8$ x 3

Order 6: $S_3$ x 14, $C_6$ x 9

Order 5: $C_5$

Order 4: $C_2^2$ x 13, $C_4$ x 9

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 51840: $F_9:S_6$

Order 25920: $A_6:\SOPlus(4,2)$, $F_9\times A_6$, $A_6:\PSU(3,2)$

Order 12960: $A_6:S_3^2$, $(C_3^2\times A_6):C_4$, $C_3^2:C_4\times A_6$

Order 8640: $F_9:S_5$

Order 6480: $C_3:S_3\times A_6$, $C_3^2:S_6$

Order 5760: $C_8:S_6$

Order 5184: $F_9:\SOPlus(4,2)$

Order 4320: $A_5\times F_9$, $A_5:\PSU(3,2)$, $A_5:\SOPlus(4,2)$, $S_3\times S_6$

Order 3456: $F_9:C_2\times S_4$

Order 3240: $C_3^2\times A_6$

Order 2880: $C_8\times A_6$, $A_6:Q_8$, $C_4:S_6$

Order 2592: $C_3^3:C_{12}:Q_8$, $C_3^2:C_4\times F_9$, $F_9:S_3^2$, $C_3^3:C_{12}:D_4$, $C_3^4:(C_2\times \SD_{16})$

Order 2160: $C_3^2:C_4\times A_5$, $A_5:S_3^2$, $(C_3\times \GL(2,4)):C_4$, $C_3:S_6$, $S_3\times A_6$, $C_3\times S_6$

Order 1728: $S_4\times \PSU(3,2)$, $D_6^2:D_6$, $A_4\times F_9:C_2$, $\PSU(3,2):S_4$, $\SOPlus(4,2).S_4$, $F_9:S_4$, $S_4\times F_9$

Order 1440: $F_9:F_5$, $C_2\times S_6$, $C_4\times A_6$, $A_6:C_4$

Order 1296: $C_3^3:(S_3\times D_4)$ x 2, $C_3^3:(S_3\times Q_8)$, $C_3^4:(C_2\times D_4)$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^2:(C_2\times F_9)$, $C_3:S_3\times F_9$, $C_3^2\wr C_2.D_4$, $C_3^2\wr C_2.D_4$, $C_3^4:C_4^2$

Order 1152: $D_4\times F_9:C_2$

Order 1080: $\GL(2,4):S_3$, $C_3^2:S_5$, $C_3\times A_6$

Order 960: $C_8:S_5$

Order 864: $C_3^2:C_4\times S_4$ x 2, $A_4:\PSU(3,2)$ x 2, $S_4\times S_3^2$, $A_4\times \PSU(3,2)$, $D_6^2:C_6$, $S_3^2:S_4$, $C_6^2:D_{12}$, $A_4\times F_9$, $A_4:F_9$, $F_9:D_6$

Order 720: $D_5\times F_9$, $D_5.\PSU(3,2)$, $D_5.\SOPlus(4,2)$, $S_3\times S_5$, $C_2\times A_6$, $S_6$

Order 648: $C_3^4:D_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3:S_3^3$, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^4:(C_2\times C_4)$, $C_3^3:(C_4\times S_3)$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^2\times F_9$

Order 576: $F_9:C_2^3$, $D_4\times \PSU(3,2)$, $C_3^2:D_4^2$, $D_4\times F_9$, $C_6^2:\SD_{16}$, $C_6^2:\SD_{16}$, $F_9:D_4$, $\PSU(3,2):D_4$, $(C_3\times C_{12}):\SD_{16}$, $F_9:D_4$, $C_4\times F_9:C_2$, $C_8:\SOPlus(4,2)$

Order 540: $C_3\times \GL(2,4)$

Order 480: $A_5:Q_8$, $C_4:S_5$, $C_8\times A_5$

Order 432: $C_6^2:C_{12}$ x 2, $(C_3^2\times A_4):C_4$ x 2, $S_3^3:C_2$ x 2, $A_4\times S_3^2$, $A_4:S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$, $S_3\times \PSU(3,2)$, $F_9:S_3$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $F_9:C_6$, $S_3\times F_9$

Order 384: $\SD_{16}\times S_4$

Order 360: $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $D_5.S_3^2$, $C_5:F_9$, $C_5\times F_9$, $S_3\times A_5$, $C_3:S_5$, $C_3\times S_5$, $A_6$

Order 324: $C_3^2:S_3^2$ x 2, $C_3\wr C_2^2$ x 2, $C_3^4:C_4$ x 2, $C_3^3:C_{12}$ x 2, $C_3^4:C_4$, $C_3\wr C_4$

Order 288: $F_9:C_2^2$ x 5, $C_3^2:C_4\times D_4$ x 2, $C_6^2:Q_8$ x 2, $C_4:\PSU(3,2)$ x 2, $C_4:\SOPlus(4,2)$ x 2, $\PSU(3,2):C_4$ x 2, $\SOPlus(4,2):C_4$ x 2, $D_4\times S_3^2$, $C_4\times \PSU(3,2)$, $C_6^2:D_4$, $D_6\wr C_2$, $S_3^2:D_4$, $C_4\times \SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $C_6^2:C_8$, $C_4:F_9$, $C_4\times F_9$, $F_9:C_4$, $C_{24}:D_6$, $C_3^2:C_4\times C_8$, $C_6^2:Q_8$, $C_6^2:D_4$, $C_2^2\times F_9$, $D_6\times S_4$

Order 240: $C_4\times A_5$, $A_5:C_4$, $C_2\times S_5$

Order 216: $C_3^2:C_4\times S_3$ x 3, $S_3^3$ x 2, $C_3^3:Q_8$ x 2, $C_3^2:D_{12}$ x 2, $S_3^2:S_3$ x 2, $S_3^2:C_6$ x 2, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2\times S_4$, $C_3^3:Q_8$, $C_3:F_9$, $C_3\times F_9$

Order 192: $Q_8:S_4$, $A_4:\SD_{16}$, $C_8:S_4$, $C_8\times S_4$, $Q_8\times S_4$, $D_4\times S_4$, $A_4\times \SD_{16}$

Order 180: $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_{15}:C_{12}$, $\GL(2,4)$

Order 162: $C_3^2\wr C_2$ x 3, $C_3^3:S_3$

Order 160: $C_{40}:C_4$

Order 144: $S_3^2:C_2^2$ x 6, $S_3\times S_4$ x 5, $F_9:C_2$ x 4, $C_2.\PSU(3,2)$ x 4, $C_2\times \PSU(3,2)$ x 3, $C_2\times F_9$ x 3, $C_3^2:C_4^2$ x 3, $C_2.\SOPlus(4,2)$ x 3, $C_6^2:C_4$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_4$ x 2, $(C_3\times C_{12}):C_4$ x 2, $S_3^2:C_4$ x 2, $C_{24}:S_3$, $C_{24}:C_6$, $C_{12}.D_6$, $D_6^2$, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $C_{12}:D_6$, $C_{12}:D_6$, $D_6:D_6$, $D_6:D_6$, $S_3\times D_{12}$, $C_4\times S_3^2$, $C_{12}.D_6$, $C_3^2:(C_2\times C_8)$, $C_3^2:\SD_{16}$

Order 128: $D_4\times \SD_{16}$

Order 120: $S_3\times F_5$, $C_2\times A_5$, $S_5$

Order 108: $C_3:S_3^2$ x 5, $C_3\times S_3^2$ x 4, $C_3^3:C_4$ x 3, $C_3^2:C_{12}$ x 3, $C_3:S_3^2$ x 2, $C_3^2\times A_4$

Order 96: $C_4\times S_4$ x 2, $A_4:Q_8$ x 2, $C_8\times A_4$, $A_4:C_8$, $C_2^2\times S_4$, $D_4\times D_6$, $Q_8\times A_4$, $D_4\times A_4$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $S_3\times \SD_{16}$

Order 90: $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 81: $C_3^4$

Order 80: $C_{20}:C_4$ x 2, $C_8\times D_5$

Order 72: $C_2\times C_3^2:C_4$ x 8, $\SOPlus(4,2)$ x 8, $S_3\times D_6$ x 5, $S_3\times A_4$ x 3, $C_3:S_4$ x 3, $C_3\times S_4$ x 3, $\PSU(3,2)$ x 3, $F_9$ x 3, $C_{12}:S_3$ x 2, $C_3\times D_{12}$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $C_6:D_6$, $C_6\times D_6$, $C_6\times A_4$, $D_4\times C_3^2$, $C_6^2:C_2$, $C_3:D_{12}$, $C_6\wr C_2$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3:D_{12}$, $C_6.D_6$, $C_3^2:C_8$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_2^2\times \SD_{16}$, $D_4\times Q_8$, $D_4^2$, $C_8:D_4$, $C_8:D_4$, $C_4:\SD_{16}$, $Q_8:D_4$, $C_2^2:\SD_{16}$, $Q_8:D_4$, $C_4\times \SD_{16}$, $C_8\times D_4$

Order 60: $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$, $A_5$

Order 54: $C_3^2:C_6$ x 5, $S_3\times C_3^2$ x 5, $C_3^2:S_3$ x 3

Order 48: $S_3\times D_4$ x 7, $C_2\times S_4$ x 5, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_{24}:C_2$, $C_2^2\times D_6$, $C_2^2\times A_4$, $C_6\times D_4$, $C_6:D_4$, $S_3\times Q_8$, $S_3\times C_8$, $C_2\times D_{12}$, $C_4\times D_6$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$, $C_{40}$, $C_5:C_8$

Order 36: $S_3^2$ x 12, $C_3^2:C_4$ x 9, $C_6\times S_3$ x 5, $C_6:S_3$ x 4, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2, $C_3\times A_4$ x 2, $C_6^2$

Order 32: $C_2\times \SD_{16}$ x 5, $C_4\times D_4$ x 3, $D_4:C_4$ x 2, $C_2^2\times D_4$ x 2, $C_4:Q_8$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $Q_8:C_4$ x 2, $C_2^2:C_8$, $C_2^2\times Q_8$, $C_2^2\times C_8$, $C_4:D_4$, $C_4\times C_8$, $C_2^2\wr C_2$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 7, $C_4\times S_3$ x 6, $C_3:D_4$ x 5, $S_4$ x 5, $D_{12}$ x 4, $C_3\times D_4$ x 4, $C_2\times A_4$ x 3, $C_3:Q_8$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_{24}$, $C_2^2\times C_6$, $C_3\times Q_8$, $C_3:C_8$

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 11, $C_3:S_3$ x 10, $C_3\times C_6$ x 3

Order 16: $C_2\times D_4$ x 10, $C_4:C_4$ x 7, $\SD_{16}$ x 4, $C_2^2:C_4$ x 4, $C_2\times C_8$ x 3, $C_2\times Q_8$ x 3, $C_2^2\times C_4$ x 3, $C_4^2$ x 2, $C_2^4$

Order 15: $C_{15}$

Order 12: $D_6$ x 12, $C_2\times C_6$ x 4, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 7

Order 8: $D_4$ x 11, $C_2\times C_4$ x 10, $C_2^3$ x 5, $Q_8$ x 3, $C_8$ x 3

Order 6: $S_3$ x 8, $C_6$ x 5

Order 5: $C_5$

Order 4: $C_2^2$ x 8, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 51840: $F_9:S_6$ ($C_1$)

Order 25920: $A_6:\SOPlus(4,2)$ ($C_2$), $F_9\times A_6$ ($C_2$), $A_6:\PSU(3,2)$ ($C_2$)

Order 12960: $C_3^2:C_4\times A_6$ ($C_2^2$)

Order 6480: $C_3:S_3\times A_6$ ($D_4$)

Order 3240: $C_3^2\times A_6$ ($\SD_{16}$)

Order 360: $A_6$ ($F_9:C_2$)

Order 72: $F_9$ ($S_6$)

Order 36: $C_3^2:C_4$ ($C_2\times S_6$)

Order 18: $C_3:S_3$ ($C_4:S_6$)

Order 9: $C_3^2$ ($C_8:S_6$)

Order 1: $C_1$ ($F_9:S_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 51840: $F_9:S_6$ ($C_1$)

Order 25920: $A_6:\SOPlus(4,2)$ ($C_2$), $F_9\times A_6$ ($C_2$), $A_6:\PSU(3,2)$ ($C_2$)

Order 12960: $C_3^2:C_4\times A_6$ ($C_2^2$)

Order 6480: $C_3:S_3\times A_6$ ($D_4$)

Order 3240: $C_3^2\times A_6$ ($\SD_{16}$)

Order 360: $A_6$ ($F_9:C_2$)

Order 72: $F_9$ ($S_6$)

Order 36: $C_3^2:C_4$ ($C_2\times S_6$)

Order 18: $C_3:S_3$ ($C_4:S_6$)

Order 9: $C_3^2$ ($C_8:S_6$)

Order 1: $C_1$ ($F_9:S_6$)

Series

Derived series

$F_9:S_6$

$\rhd$

$C_3^2:C_4\times A_6$

$\rhd$

$C_3^2\times A_6$

$\rhd$

$A_6$

Chief series

$F_9:S_6$

$\rhd$

$F_9\times A_6$

$\rhd$

$F_9$

$\rhd$

$C_3^2:C_4$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$F_9:S_6$

$\rhd$

$C_3^2:C_4\times A_6$

$\rhd$

$C_3:S_3\times A_6$

$\rhd$

$C_3^2\times A_6$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $54 \times 54$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.